Working on fairly unrelated stuff, I needed to prove the following, fairly easy results, and I wonder if anyone can provide references to the literature. Not being a probabilist I wouldn't know where to start looking.

I. Let $X$ and $Y$ be two integrable random variables, and let $X'$ and $Y'$ have the same distributions, respectively, and be indepdendent from $X$ and $Y$. Then $$ E[\min \{X,X'\} + \min \{Y,Y'\} - 2\min\{ X,Y'\}] \geq 0 $$ and equality holds iff $X$ and $Y$ have the same distribution

II. Let $X$ be a random variable, and let $\cal F$ be some sub-algebra. Let $Y$ have the same conditional distribution as $X$ over $\cal F$, indepndently from $X$ over $\cal F$, and let $Z$ have the same distribution as $X$, independently from $X$. Then $$ E[\min \{X,Y\}] \geq E[ \min \{X,Z\} ] $$ and equality holds iff $X$ is independent from $\cal F$.